Classification of finite group automorphisms with a large cycle

Let $\psi$ be a permutation of a finite set $X$. We define $\lambda(\psi)$ to be the largest fraction of elements of $X$ lying on a single cycle of $\psi$. For a finite group $G$, we define $\lambda(G)$ to be the maximum among the values $\lambda(\alpha)$, where $\alpha$ runs through the automorphisms of $G$. In this paper, we develop tools to deal with questions related to $\lambda$-values of finite groups and of their automorphisms. As a consequence, we will be able to give a classification, up to a natural notion of isomorphism, of those pairs $(G,\alpha)$ where $G$ is a finite group, $\alpha$ is an automorphism of $G$ and $\lambda(\alpha)\geq\frac{1}{2}$.